Given = 1 + 1z-141 -1z-11 + 1z-123For three different ROC,

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Given = 1 + 1 z^-1
4
1 - 1 z^-1 1 + 1 z^-1
2 3

For three different ROC, consider three different solution of signal x[n].
1. |z| > 1 , x[n] 1 - -1 ⁿ u[n]
2 2^n-1 3

2. |z| < 1 , x[n] 1 - -1 ⁿ u[-n+1]
3 2^n-1 3

3. 1 < |z| < 1 , x[n] = 1 u[- n - 1] - -1 ⁿ u[n]
3 2 2^n-1 3

Of these the correct solutions are

1. 1 and 2
2. 1 and 3
3. 2 and 3
4. 1, 2 and 3

Correct Option: B

X(z) =	1				+			- 1				\|z\| > 1/2 (Right-Sidesd)

	1 -	1	z^-1					1 +	1	z^-1
		2							z

⇒ x[n] =	2	u[n] -		-1		ⁿ	u[n] \|z\| < 1/3 (Left-Sided)
	2ⁿ			3

⇒ x[n] =		- 2	+		-1		ⁿ		u[-n-1] 1/3 < \|Z\| < 1/2(Two-Sided)
		2ⁿ			3

x[n] = -	2	u[-n-1] -		-1		ⁿ	u[-n+1]
	2ⁿ			3